3.2 General multi-hole solutions

Time-symmetric black-hole solutions of the constraint equations such as those described in Section 3.1
are useful as test cases because they have analytic representations. However, they have very little physical
relevance. General time-asymmetric solutions are needed to represent black holes that are moving and
spinning9.
A few approaches for generating general multi-hole solutions have been explored and, below, we will look at
the two approaches which are direct generalizations of the Misner and Brill–Lindquist data. Generalizing
these two approaches was the most natural first step toward constructing general multi-hole
solutions.

The first approach to be developed generalized the Misner approach [79]. It was attractive because an
isometry condition relating the two asymptotically flat universes provides two useful things. First, because
the two universes are identical, finding a solution in one universe means that you have the full solution.
Second, because the throats are fixed-point sets of the isometry, we can construct boundary
conditions on any quantity there. This allows us to excise the region interior to the spherical throats
from one of the Euclidean background spaces and solve for the initial data in the remaining
volume.

The generalization of the Misner approach seemed preferable to trying to solve the constraints on
Euclidean manifolds stitched together smoothly at the throats of black holes. However,
Brandt and Brügmann [27] realized it was possible to factor out analytically the behavior of the
singular points in the Euclidean manifold of the sheeted approach. Referred to as the
“puncture” method, this approach allows us to rewrite the constraint equations for functions on an
sheeted manifold as constraint equations for new functions on a simple Euclidean
manifold.

Another approach tried, which we will not discuss in detail, avoided the issue of the topology of the
initial-data slice entirely. Developed by Thornburg [100], this approach was based on the idea that only the
domain exterior to the apparent horizon of a black hole is relevant. The equation describing the location of
an apparent horizon can be rewritten in a form that can be used as a boundary condition for the conformal
factor in the Hamiltonian constraint equation. Thus, given a compatible solution to the momentum
constraints, this boundary condition can be used to construct a solution of the Hamiltonian constraint in
the domain exterior to the apparent horizons of any black holes, with no reference at all to the topology of
the full manifold.

3.2.1 Bowen–York data

The generalization of the Misner approach was developed by York and his collaborators [25, 23, 24, 67, 113, 66, 26, 38].
The approach is often called the “conformal-imaging” method, and the data are usually referred to as
“Bowen–York” data. This approach begins with a set of simplifying assumptions that is common to all
three of the approaches described above. These assumptions are that

Here, represents a flat metric in any suitable coordinate system. The assumption of conformal flatness
means that the differential operators in the constraints are the familiar flat-space operators. More
importantly, if we use the conformal transverse-traceless decomposition (32), we find that in vacuum the
momentum constraints completely decouple from the Hamiltonian constraint.

The importance of this last property stems from the fact that York and Bowen were able to find analytic
solutions of this version of the momentum constraints, solutions that represent a black hole with both linear
momentum and spin [25, 23, 24]. If we choose , then the momentum constraints (31) become

A solution of this equation is

Here, and are vector parameters, is a coordinate radius, and is the outward-pointing
unit normal of a sphere in the flat conformal space (). is the 3-dimensional Levi-Civita
tensor.

This solution of the momentum constraints yields the tracefree part of the extrinsic curvature,

Remarkably, using this solution (70) and the assumptions in (67), we can determine the physical values for
the linear and angular momentum of any initial data we can construct. The momentum contained
in an asymptotically flat initial-data hypersurface can be calculated from the integral [112]

where is a Killing vector of the 3-metric
10.
Since we are not likely to have true Killing vectors, we make use of the asymptotic translational and
rotational Killing vectors of the flat conformal space. We find from (71), (70), and (67) that the physical
linear momentum of the initial-data hypersurface is and the physical angular momentum of the slice is
. Furthermore, because the momentum constraints are linear, we can add any number of solutions of
the form of Eq. (70) to represent a collection of linear and angular momentum sources. The total physical
linear momentum of the initial-data slice will simply be the vector sum of the individual linear momenta.
The total physical angular momentum cannot be obtained by simply summing the individual
spins because this neglects the orbital angular momentum of the various sources. However,
the total angular momentum can still be computed without having to solve the Hamiltonian
constraint [114].

The Bowen-York solution for the extrinsic curvature is the starting point for all the general multi-hole
initial-data sets we have discussed in Section 3.2. However, this solution is not inversion symmetric. That
is, it does not satisfy the isometry condition that any field must satisfy to exist on a two-sheeted
manifold like that of Misner’s solution. Fortunately, there is a method of images, similar to that
used in electrostatics but applicable to tensors, that can be used to make any tensor inversion
symmetric [79, 25, 67, 66, 113].

For the conformal extrinsic curvature of a single black hole, there are two inversion-symmetric
solutions [25],

Here, is the radius of the coordinate 2-sphere that is the throat of the black hole. Of course, this
coordinate 2-sphere is the fixed-point set of the isometry and is the surface on which we can impose
boundary conditions. Notice that this radius enters the solutions only when we make it inversion
symmetric.

When the extrinsic curvature represents more than one black hole, the process for making the solution
inversion symmetric is rather complex and results in an infinite-series solution. However, in most cases
of interest, the solution converges rapidly and it is straightforward to evaluate the solution
numerically [38].

Given an inversion-symmetric conformal extrinsic curvature, it is possible to find an inversion-symmetric
solution of the Hamiltonian constraint [25]. Given our assumptions (67), the Hamiltonian constraint
becomes

The isometry condition imposes a condition on the conformal factor at the throat of each hole. This
condition takes the form [25]

where is the outward-pointing unit-normal vector to the throat and is the coordinate radius
of that throat. This condition can be used as a boundary condition when solving (73) in the region exterior
to the throats.

In addition to boundary conditions on the throats, a boundary condition on the outer boundary of the
domain is needed before the quasilinear elliptic equation in (73) can be solved as a well-posed
boundary-value problem. This final boundary condition comes from the fact that we want an asymptotically
flat solution. This implies that the solution behaves as

where is the total ADM energy content of the initial-data hypersurface. Equation (75) can be used to
construct appropriate boundary conditions either at infinity or at a large, but finite, distance from the black
holes [116].

3.2.2 Puncture data

The generalization of the Brill–Lindquist data developed by Brandt and Brügmann [27] begins with the
same set of assumptions (67) as the conformal-imaging approach outlined in Section 3.2.1. We immediately
have Eq. (70) from the solution of the momentum constraints, and we must solve the Hamiltonian
constraint, which again takes the form of Eq. (73). At this point, however, the method of solution differs
from the conformal-imaging approach.

Based on the time-symmetric solution, it is reasonable to assume that the conformal factor will take the
form

If is sufficiently smooth, (76) implies that the manifold will have the topology of
asymptotically flat regions just as in the Brill–Lindquist solution. In this case, asymptotic flatness requires
that .

Near each singular point, or “puncture”, we find that . From (70), we see that
behaves no worse than , so vanishes at the punctures at least as fast as
.

With this behavior, Brandt and Brügmann [27] have shown the existence and uniqueness of
solutions of the modified Hamiltonian constraint (77). The resulting scheme for constructing multiple black
hole initial data is very simple. The mass and position of each black hole are parameterized
by and , respectively. Their linear momenta and spin are parameterized by
and in the conformal extrinsic curvature (70) used for each hole. Finally, the solution for
is found on a simple Euclidean manifold, with no need for any inner boundaries to avoid
singularities. This is a great simplification over the conformal-imaging approach, where proper
handling of the inner boundary is the most difficult aspect of solving the Hamiltonian constraint
numerically [41].

3.2.3 Problems with these data

Both the conformal-imaging and puncture methods for generating multiple black hole initial data allow for
completely general configurations of the relative sizes of the black holes, as well as their linear and angular
momenta. This does not mean that these schemes allow for the generation of all desired black-hole initial
data. The two schemes rely on specific assumptions about the freely specifiable gravitational data. In
particular, they assume , , and, most importantly, that the 3-geometry is conformally
flat.

These choices for the freely specifiable data are not always commensurate with the desired physical
solution. For example, if we choose to use either method to construct a single spinning black hole, we will
not obtain the Kerr solution. The Kerr–Newman solution can be written in terms of a quasi-isotropic radial
coordinate on a time slice [28]. Let denote the usual Boyer–Lindquist radial coordinate and
make the standard definitions

A quasi-isotropic radial coordinate can be defined via

The interval then becomes

with

We see immediately that the 3-geometry associated with a hypersurface of (81) is not
conformally flat. In fact, Garat and Price [54] have shown that in general there is no spatial slicing of the
Kerr spacetime that is axisymmetric, conformally flat, and smoothly goes to the Schwarzschild solution as
the spin parameter .

Since the Kerr solution is stationary, the inescapable conclusion is that conformally flat initial data for a
single rotating black hole must also contain some nonvanishing dynamical component. When we evolve
such data, the system will emit gravitational radiation and eventually settle down to the Kerr
geometry [25, 29]. But, it cannot be the Kerr geometry initially, and it is unlikely that the
spurious gravitational radiation content of the initial data has any desirable physical properties.
Conformally flat initial data for spinning holes contain some amount of unphysical “junk” radiation.
A similar conclusion is reached for conformally flat data for a single black hole with linear
momentum [112].

The choice of a conformally flat 3-geometry was originally made for convenience. Combined with the
choice of maximal slicing, these simplifying assumptions allowed for an analytic solution of the momentum
constraints which vastly simplified the process of constructing black-hole initial data. Yet there has been
much concern about the possible adverse physical effects that these choices (especially the choice of
conformal flatness) will have in trying to study black-hole spacetimes [40, 71, 88, 61, 87]. While these
conformally flat data sets may still be useful for tests of black-hole evolution codes, it is becoming widely
accepted that the unphysical initial radiation will significantly contaminate any gravitational
waveforms extracted from evolutions of these data. In short, these data are not astrophysically
realistic.

The various initial-data decompositions outlined in Section 2.2 and Section 2.3 are all capable of
producing completely general black-hole initial data sets. The only limitation of these schemes is our
understanding of what choices to make for the freely specifiable data and the boundary conditions to apply
when solving the sets of elliptic equations. All these choices will have a critical impact on the astrophysical
significance of the data produced. It is also important to remember that similar choices for the
freely specifiable data will result in physically different solutions when applied to the different
schemes.

The first studies of black-hole initial data that are not conformally flat were carried out by
Abrahams et al.[1]. They looked at the superposition of a gravitational wave and a black hole.
Using a form of the conformal metric that allows for so-called Brill waves, they constructed
time-symmetric initial data that were not conformally flat and yet satisfied the isometry condition (65)
used in the conformal-imaging method of Section 3.2.1. These data were further generalized by
Brandt and Seidel [30] to include rotating black holes with a superimposed gravitational wave. In
this case, the data are no longer time-symmetric yet they satisfy a generalized form of the
isometry condition so that the solution is still represented on two isometric, asymptotically flat
hypersurfaces.

Matzner et al.[78] have begun to move beyond conformally flat initial data for binary black holes Their
proposal is to use boosted versions of the Kerr metric written in the Kerr–Schild form to represent each
black hole. Thus, an isolated black hole will have no spurious radiation content in the initial
data. To construct solutions with multiple black holes, they propose, essentially, to use a linear
combination of the single-hole solutions. The resulting metric can be used as the conformal 3-metric,
the trace of the resulting extrinsic curvature can be used for , and the tracefree part of
the resulting extrinsic curvature can be used for . Their scheme uses York’s conformal
transverse-traceless decomposition outlined in Section 2.2.1, with the boundary conditions
of and on the horizons of the black holes and conditions appropriate for
asymptotic flatness at large distances from the holes. A related method has been proposed by
Bishop et al.[15], but their approach is much different and outside the current scope of this
review.

The approach outlined by Matzner should certainly yield “cleaner” data than the conformally flat data
currently available. For the task of specifying data for astrophysical black-hole binaries in nearly circular
orbits, it is still true that these new data will not contain the correct initial gravitational wave
content. Because the black holes are in orbit, they must be producing a continuous wave-train of
gravitational radiation. This radiation will not be included in the method proposed by Matzner etal. Also, it is clear that the boundary conditions being used do not correctly account for the
tidal distortion of each black hole by its companion. When the black holes are sufficiently far
apart, the radiation from the orbital motion can be computed using post-Newtonian techniques.
One possibility for producing astrophysically realistic, binary black-hole initial data is to use
information from these post-Newtonian calculations to obtain better guesses for , , and
.